An even better Density Increment Theorem and its application to Hadwiger's Conjecture

Abstract

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t{)}^{\beta })$-colorable for every $\beta >1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt{\log t})$ bound. More recently, the author showed that every graph with no $K_t$ minor is $O(t(\log t{)}^{\beta })$-colorable for every $\beta >0$; more specifically, they are $t\cdot 2^{O((\log \log t{)}^{2/3})}$-colorable. In combination with that work, we show in this paper that every graph with no $K_t$ minor is $O(t(\log \log t{)}^6)$-colorable.

Obsolete (for the purposes of improved bounds on Hadwiger's Conjecture) due to the stronger result (and shorter proof) by the author and Delcourt in arXiv:2108.01633

References in corpus (3)

• Further progress towards Hadwiger's conjecture
• A new upper bound on the chromatic number of graphs with no odd $K_t$ minor
• Connectivity and choosability of graphs with no $K_t$ minor

Cited by in corpus (6)

• Further progress towards Hadwiger's conjecture
• Further Progress towards the List and Odd Versions of Hadwiger's Conjecture
• Immersion and clustered coloring
• Clique minors in graphs with a forbidden subgraph
• Complete minors in digraphs with given dichromatic number
• Improved bound for Hadwiger's conjecture